energies

Article New Adaptive High Starting Scalar Control Scheme for Induction Motors Based on Passivity

Juan Carlos Travieso-Torres 1,* , Miriam Vilaragut-Llanes 2 , Ángel Costa-Montiel 2, Manuel A. Duarte-Mermoud 3 , Norelys Aguila-Camacho 4 , Camilo Contreras-Jara 1 and Alejandro Álvarez-Gracia 1

1 Department of Industrial Technologies, University of Santiago de Chile, Santiago 9170125, Chile; [email protected] (C.C.-J.); [email protected] (A.Á.-G.) 2 Centro de Investigaciones y Pruebas Electroenergéticas (CIPEL), Universidad Tecnológica de la Habana “José Antonio Echeverría” (CUJAE), Havana 11500, Cuba; [email protected] (M.V.-L.); [email protected] (Á.C.-M.) 3 Departamento de Electricidad, University of Chile, Santiago 8370451, Chile; [email protected] 4 Departamento de Electricidad, Universidad Tecnológica Metropolitana, Santiago 7800002, Chile; [email protected] * Correspondence: [email protected]

 Received: 14 December 2019; Accepted: 19 January 2020; Published: 10 March 2020 

Abstract: A novel adaptive high starting torque (HST) scalar control scheme (SCS) for induction motors (IM) is proposed in this paper. It uses a new adaptive-passivity-based controller (APBC) proposed herein for a class of nonlinear systems, with linear explicit parametric dependence and linear stable internal dynamics, which encompasses the IM dynamical model. The main advantage of the HST-SCS includes the ability to move loads with starting-torque over the nominal torque with a simple and cost-effective implementation without needing a speed sensor, variable observers, or parameter estimators. The proposed APBC is based on a direct control scheme using a normalized fixed gain (FG) to fine-tune the adaptive controller parameters. The basic SCS for induction motors (IM) and the HST-SCS were applied to an IM of 200 HP and tested using a real-time simulator controller OPAL-RT showing the achievement of the proposal goal.

Keywords: adaptive-passivity-based controller (APBC); scalar control scheme (SCS); (IM); proportional (P) controller; nonlinear dynamical systems

1. Introduction Compared to electric motors, induction motors (IMs) have a lower cost and higher efficiency, require lower maintenance, and have been replacing them in variable speed operations with increasing use in the past twenty years [1]. In variable speed applications, the IM is fed by different types of (AC) drives [2], which differ in their performance with respect to the starting torque, transient speed behavior, and steady-state speed-accuracy of the desired speed [1]. For low, medium, and high-performance applications, three main control schemes are used: scalar control scheme (SCS) [3], direct torque control (DTC) [4], and field-oriented control (FOC) [5,6] respectively. There is also a simpler AC/AC soft based on thyristors [3], but it does not allow variable speed operation. It only reduces the starting current while moving low starting torque loads. This reduction is made by reducing the starting applied to the motor. The AC drive with a rotor speed closed loop (CL), based on FOC or DTC, is used for high-performance applications, such as winding and takeoffs. Its capabilities consist of starting with torque around 150% of the nominal torque, having a rapid and no oscillatory transient speed

Energies 2020, 13, 1276; doi:10.3390/en13051276 www.mdpi.com/journal/energies Energies 2020, 13, 1276 2 of 15 behavior, and exhibiting 0.01% of steady-state speed accuracy. It needs parameter estimators to compute its characteristic variable slip frequency of operation. The AC drive based on sensorless DTC or sensorless FOC, is used for medium-performance applications. These consist of cranes, positive displacement pumps, and compressors. Its capabilities consist of starting with torque around 120% of the nominal torque, having a rapid and no oscillatory transient speed behavior, and 0.1% of steady-state speed accuracy. It needs parameter estimators and variable observers to compute its characteristic direct torque and flux control. In contrast, the cheapest AC-drive-based SCS is used for low-performance applications. Examples of these include blowers, fans, and centrifugal pumps. SCS only needs the rated voltage per phase and the frequency taken from the motor data plate. It moves loads needing a starting torque lower than 40% of the nominal torque [7] with 1% to 3% of steady-state speed accuracy and nonrapid and even oscillatory transient behavior [8–10]. These facts about SCS remain even if a CL speed control is considered as mainly improving steady-state accuracy [9]. This is the reason why FOC is finally proposed in [10] to reach a higher performance. Oscillatory transient behavior issues of the SCS are studied in detail in [11]. Several research works aim to improve the simplest SCS [12–17]. Compared to the DTC [4] and the FOC [5,6], the SCS needs neither parameter estimators nor estimator or sensor of the rotor speed. SCS schemes [3,12–17] assume that the electrical angular speed is equal to the reference rotor angular speed, neglecting the slip, or equals to the reference rotor angular speed plus an estimated slip. These are the simplest and most effective ways the estimated rotor angle is implemented. Improvements in works [12–15] consider estimating the slip to improve the rotor steady-state speed-accuracy. The electromagnetic torque is estimated in [12], and later the slip is based on this estimate. The gap power is estimated in [13] and then the slip is estimated, adjusting also the boost voltage. The slip is calculated based on a stator flux observer in [14], which depends on parameter estimators. A simpler slip calculation scheme is proposed in [15], where the computing depends on the rated stator current and the rated slip. The speed control of all these SCS schemes has an open-loop (OL) speed control. In addition, to improve the steady-state rotor speed accuracy of the SCS, a CL speed controller is proposed in [16]. The required electrical frequency is set by a CL speed fuzzy controller in [16]. It depends on the measurement of a rotor angular speed sensor. A rotor angular speed observer based on a spectral search method is proposed in [17] to improve the SCS speed monitoring. It is suggested in [17] that this monitoring could be used in other works to develop a CL sensorless SCS, but this is not fully addressed by [17]. All these schemes [12–17] aim to improve the accuracy of the steady-state rotor speed. They all need parameter estimators, depending on the accuracy of the estimates. The low starting torque issue of the SCS is not addressed in these references from [12–17]. It is in this paper where a solution is suggested. The mining and minerals industry, for instance, has machinery such as conveyor belts. These need high starting torque but no steady-state speed accuracy. One employed solution is to consider AC drives with a DTC or FOC scheme, paying an over cost. Another solution considers an AC drive with SCS to move the belt conveyors unloaded at the start. This last solution has an issue when unforeseen detentions occur. This paper aims to propose an alternative simple solution to move high starting-torque loads. It uses for current-control purposes the same current sensors used today by the basic SCS for motor protection. Compared to the DTC [4] and the FOC [5,6], the proposed high starting torque (HST)-SCS needs neither parameter estimators, variable observers, nor estimator or sensor of the rotor speed. As a result, it moves loads needing a starting torque around 100% of the nominal torque from 1% to 3% of steady-state speed accuracy and nonrapid and even oscillatory transient behavior [8–10]. It expands the applications of the SCS but is not capable of substituting the FOC nor DTC schemes. The main contribution of this paper is the proposal of an HST-SCS. It adds a CL adaptive-passivity-based controller (APBC) for current to the basic SCS. This is based on the adaptive theory from [18,19] developed for linear dynamical systems but extended herein for a class of nonlinear Energies 2020, 13, 1276 3 of 15 dynamical systems expressed in the normal form, which has a linear explicit parametric dependence with a linear internal dynamic and encompasses the IM model. The use of normalized FG is proposed to adjust the adaptive controller parameters. The paper is organized as follows: Section2 describes the IM model, the basic SCS method, and the APBC basis. The proposed APBC method for a class of nonlinear systems is proposed in Section3. Experimental results comparing basic SCS and the proposed HST-SCS are described in Section4. Finally, in Section5, some conclusions are drawn.

2. Preliminaries The design and performance of the proposed HST-SCS for IM will be compared to the basic SCS for the IM described in this section. Both schemes make use of several concepts and theories also described in what follows.

2.1. Basic SCS for IM The steady-state IM-equivalent circuit per phase is described in detail in Appendix A.1[ 20]. The basic SCS considers a P controller to keep constant magnetization at different electrical frequencies as shown in the following equation:     √2p Vs_rated Vboost  P1ωe∗ + √2Vboost, with P1 = for fmin < f < fc,  2π frated − f c Vs∗ =   (1)  √2p Vs_rated  P2ω∗, with P2 = for fc < f < frated,  e 2π frated where Vs∗ is the required amplitude of the phase stator voltage to be applied in order to achieve the required electrical angular frequency ωe∗. In practice, ωe∗ is considered to be equal to the required rotor angular speed multiplied by the pair of poles ωr∗p, neglecting the rotor angular slip. Vs_rated is the rated phase voltage from the motor data plate, p is the pair of poles, frated is the motor rated frequency in Hz, and the Vboost has a value of up to 25% of Vs_rated in [21] and operates from the minimum frequency fmin (with a value up to 6% of frated in [21]) to the low-frequency fc (with a value up to 40% of frated in [21]). The Vboost is a controller bias or offset needed to deliver a certain amount of starting torque according to Equation (A4): pV 2 Rrωslip T = 3 s (2) em  2 2 ωe 2 Rr + ωslipL0r

The scalar control aims to keep a constant flux modulus ( ϕ ), neglecting the stator voltage drop in the stator impedance (Rsis + jωeL is 0). After that, the Kirchhoff’s voltage law equation for the s0 ≈ stator (A1) was simplified and shows the dependence between the flux modulus ( ϕ ), the frequency ωe, and the modulus of the voltage applied to the motor, according to

Vs 0 ϕ z}|{ z }|≈ { z}|{ us us = (Rsis + jωeLs0is) + jωe Lmim => us jωeϕ => ϕ | | . (3) ≈ ∼ ωe

Thus, the amplitude of the rated voltage per phase and the rated electrical angular frequency are 2p   √ Vs_rated related as P2 = 2π f rated = ϕrated , which is kept constant over fc throughout Equation (1). From this result, it can also be shown that at low speed, the required voltage Vs∗ = P2ωe∗ is low, and the stator impedance is no longer neglectable. This is the reason why a Vboost is applied under fc through the Equation (1) to assure the existence of flux and torque in this case. Finally, when operating at speeds higher than the rated speed ωe_rated, a saturator needs to be used after the P controller. It limits the stator voltage to the safe limit Vs_rated, thus protecting the IM. Energies 2020, 13, 1276 4 of 15

jρ After applying the Park transformation e− [22], the two-phase α–β instantaneous signals fixed at a stationary reference frame can be obtained from the d–q amplitude magnitudes rotating at the = jρ synchronous reference frame; thus, us∗ β e− Vsdq∗ . Similarly, from the α–β instantaneous signals, the ∝ jρ d–q amplitude magnitudes can be obtained through Isdq = e− is β. ∝ In addition, using the Clarke transformation T2 3 [23] from the two-phase α–β instantaneous → signals fixed at the stationary reference frame, the three-phasic instantaneous signals can be obtained Energies 2020, 13,h x FOR PEER REVIEWiT h i 4 of 14 through us = ua ub uc = T2 3u∗ and vice versa isαβ = isα isβ = T3 2is. → s β → TheThe generalgeneral diagramdiagram ofof thethe basicbasic ∝ SCSSCS forfor IMIM thatthat willwill bebe usedused inin thisthis paperpaper forfor comparisoncomparison purposespurposes isis shownshown inin FigureFigure1 1,, taken taken from from [ [21].21]. ItIt doesdoes notnot makemake useuse ofof parameterparameter estimatorsestimators whenwhen comparedcompared toto DTCDTC [[4]4] oror FOCFOC [[5,6],5,6], usingusing onlyonly thethe ratedrated voltagevoltage perper phasephase fromfrom thethe motormotor datadata plateplate andand sensorssensors of of the the stator stator current current consumption consumption to checkto check the the proper proper IM functioningIM functioning for itsfor protection. its protection. The ∗ setpointThe setpoint of the of required the required rotor angular rotor angular speed ω r∗speedis established 𝜔 is established by the operator, by the and operator, the SCS automaticallyand the SCS ∗ definesautomatically the required defines stator the required voltage V statorsd∗ and voltage electrical 𝑉 frequency and electricalωe. frequency 𝜔.

FigureFigure 1.1. Simulink block diagram ofof basicbasic SCSSCS forfor thethe inductioninduction motormotor (IM).(IM).

TheThe SCSSCS schemescheme describeddescribed inin FigureFigure1 1 is is simple simple and and e effective.ffective. ItIt isis usedused forfor applicationsapplications needingneeding lowlow startingstarting torque torque and and a steady-statea steady-state speed speed accuracy accuracy up up to 3%.to 3%. To operatesTo operates motor motor loads loads needing needing HST, someHST, issuessome issues are faced are whichfaced which are described are described in the in next the section. next section. 2.2. Problem Statement 2.2. Problem Statement After applying the basic SCS for IM, the following issues are faced: After applying the basic SCS for IM, the following issues are faced: (a)(a) HST: To keepkeep thethe constantconstant fluxflux givengiven inin (3),(3), basicbasic SCSSCS usesuses thethe controllercontroller (1),(1), whichwhich hashas aa lowlow starting torque issue. This This is is due to the low appliedapplied startingstarting voltage,voltage, eveneven afterafter applyingapplying thethe Vboost, because the torque (2) is proportional to the square of the quotient between the phase rated 𝑉, because the torque (2) is proportional to the square of the quotient between the phase ratedstator stator voltage voltage and rated and electrical rated electrical frequency. frequenc Thus,y. an Thus, alternative an alternative method controllingmethod controlling the required the stator current is∗ to have HST∗ is proposed next. required stator current 𝑖 to have HST is proposed next. (b) Nonlinear IM dynamical model with unknown parameters: To control the stator current i ∗, the (b) Nonlinear IM dynamical model with unknown parameters: To control the stator current 𝑖s∗ , the IM d-q dynamical model will be used (A5), whichwhich isis aa nonlinearnonlinear dynamicaldynamical systemsystem withwith linearlinear explicit parametric dependence and linear stable internal dynamics ((zz)) ofof thethe formform . y = A1 f1(y) + A2 f2(z) + Bu 𝑦 = 𝐴𝑓(𝑦) + 𝐴𝑓(𝑧) +𝐵𝑢 . (4) z = Cz + Dy (4) 𝑧 =𝐶𝑧+𝐷𝑦 where the flux, which is the internal dynamics variable 𝑧=𝛹 𝛹 ∈ℜ , is not accessible, the current output variable 𝑦=𝐼 𝐼 ∈ℜ and the voltage input (control) variable 𝑢= 𝑉 𝑉 ∈ℜ are both accessible. The system function 𝑓(𝑦) = 𝑦 𝜔𝑦 ∈ℜ is known, but 𝑓 (𝑧) =𝑧 𝜔 𝑧 ∈ℜ the function is unknown. All system parameters are considered constant ´ 01 × but unknown, where A =− 𝐼 ∈ℜ , 𝐼 is the identity matrix of order 2, and A = −1 0 − 𝜔 01 × × × 𝐼 ∈ℜ , 𝐵 = 𝐼 ∈ℜ , 𝐶= ∈ℜ and 𝐷= 𝐼 ∈ −1 0 −𝜔 − ℜ × . B and C are invertible matrices. The sign(B) matrix contains the sign of each element of B and

Energies 2020, 13, 1276 5 of 15

T h i 2 where the flux, which is the internal dynamics variable z = Ψrd Ψrq , is not accessible, T ∈ < h i 2 the current output variable y = Isd Isq and the voltage input (control) variable T ∈ < T h i 2 h T T i 4 u = Vsd Vsq are both accessible. The system function f1(y) = y ωe y ∈ < T ∈ < h T p T i 4 is known, but the function f2(z) = z 2 ωrz is unknown. All system parameters ∈" < " ## Rs0 0 1 4 2 are considered constant but unknown, where A = I2 × , I2 is the 1 − σLs 1 0 ∈ < " " ## − RrLm Lm 0 1 4 2 1 2 2 identity matrix of order 2, and A2 = 2 I2 σL L × , B = σL I2 × , σLsLr s r 1 0 ∈ < s ∈ <   − Rr ω  Lr slip  2 2 RrLm 2 2 C =  − R  × and D = I2 × . B and C are invertible matrices. The  ω r  ∈ < Lr ∈ < − slip − Lr sign(B) matrix contains the sign of each element of B and is assumed to be known. Therefore, an adaptive controller designed for nonlinear systems of the form (4) is proposed herein. (c) Parameters adjustment of APBC: How to adjust the controller parameters K and the fixed-gain Γ [19] remains an open question today. We will try to answer it herein by considering a known system time constant and by including a normalized FG for the APBC.

3. HST-SCS Proposal As previously described, it can be seen from (3) that the flux modulus ϕ is proportional to the | | quotient of the phase rated stator voltage and the rated frequency (after neglecting the stator voltage drop in the stator impedance). However, from (3), it can also be seen that the flux modulus ϕ is | | proportional to the modulus of the magnetizing current im = is ir. Replacing ir from (A2), we get −

 Rr   + jLr0  jLm  ωslip  im = is is = is  (5) Rr  Rr  − + jLr  + jLr  ωslip ωslip

Considering Equation (5), we explore how to keep a constant im by keeping a constant stator phase 2

= p jLm Rr 2 current is to guarantee an HST. This is obtained from (A3) and results as Tem 3 2 Rr ω is . +jLr slip ωslip | |

3.1. HST-SCS for IM A general diagram of the proposed HST-SCS is shown in Figure2. Here, a new HST control loop (marked in yellow) operating from the minimum frequency fmin to the lower frequency fc1 was added to the basic SCS. It moves the Vboost operation from fc1 to fc, keeping the P2ωe∗ operating from fc to frated. The new transition frequency fc1 from HST control to Vboost control is lower than fc. Energies 2020, 13, x FOR PEER REVIEW 5 of 14 is assumed to be known. Therefore, an adaptive controller designed for nonlinear systems of the form (4) is proposed herein. (c) Parameters adjustment of APBC: How to adjust the controller parameters K and the fixed-gain Γ [19] remains an open question today. We will try to answer it herein by considering a known system time constant and by including a normalized FG for the APBC.

3. HST-SCS Proposal As previously described, it can be seen from (3) that the flux modulus |φ| is proportional to the quotient of the phase rated stator voltage and the rated frequency (after neglecting the stator voltage drop in the stator impedance). However, from (3), it can also be seen that the flux modulus |φ| is proportional to the modulus of the magnetizing current 𝑖 =𝑖 −𝑖. Replacing 𝑖 from (A2), we get

´ 𝑖 =𝑖 − 𝑖 =𝑖 (5)

Considering Equation (5), we explore how to keep a constant 𝑖 by keeping a constant stator phase current 𝑖 to guarantee an HST. This is obtained from (A3) and results as 𝑇 = 3 |𝑖| .

3.1. HST-SCS for IM A general diagram of the proposed HST-SCS is shown in Figure 2. Here, a new HST control loop

(marked in yellow) operating from the minimum frequency 𝑓 to the lower frequency 𝑓 was ∗ added to the basic SCS. It moves the 𝑉 operation from 𝑓 to 𝑓, keeping the 𝑃𝜔 operating 𝑓 𝑓 𝑓 𝑉 Energiesfrom 2020 to, 13, 1276. The new transition frequency from HST control to control is lower6 than of 15 𝑓.

Figure 2. Simulink block diagram of the proposed HST-SCS forfor thethe IM.IM.

InIn thethe proposedproposed HST-SCS,HST-SCS, EquationEquation (1)(1) takestakes thethe formform  .  θTω, with θ = e(Γω)T for f < f < f ,  min c1  √  V   2p srated Vboost V =  P1ωe∗ + √2Vboost, with P1 = for fc1 < f < fc, (6) s∗  2π frated − f c   V   √2p srated  P2ω∗, with P2 = for fc < f < frated, e 2π frated where the added HST control loop θ is the adaptive HST controller parameter, depending on the fixed-gains Γ and the information vector ω. θ, Γ, and ω are all defined in Section 3.2.

3.2. Normalized FG-APBC The APBC basis for tracking and regulation purposes is thoroughly described in [19] for nonlinear dynamical systems with unknown parameters. It will be extended in this paper for the class of nonlinear dynamical systems (4) and using a normalized FG. Due to the characteristics of the internal dynamics z of Equation (4), Laplace transform [24] can be applied. Considering constant D and C, and assuming initial condition z(0), z(s) = 1 1 1 (sI C)− Ds− Y(s) + (s C)− z(0) is obtained. Applying the final value theorem [24], i.e., limz(t) = − − t 1 1 →∞ limsz(s), we have limz = C− DY(s). Thus, after replacing z = C− DY(s) in the terms f2(z) of Equation s 0 t → →∞ (4), we get that A1 f1(y) + A2 f2(z) = A f (y), and the dynamical system (4) takes the form

. y = A f (y) + Bu . (7) z = Cz + Dy where the output variable y n and the input (control) variable u n are accessible. All system ∈ < ∈ < parameters A n m and B n n are unknown with B invertible, and τ is the known time constant ∈ < × ∈ < × of the system (7). It is assumed that sign(B) is known. The system function f (y) m is known. ∈ < T h T T p T i 6 For the IM case, in Equation (7), the system function f (y) = y ωe y 2 ωr y , the  ∈ <   Rr  1 " # " # Rr  1  R R2L2  ωslip − 0 1 R L2 0 1  ωslip −  A =  s0 + r m  Lr  r m  Lr  , system parameters are  σL I2 3  − Rr  2  − Rr    − s σLsLr  ω  1 0 σLsLr 1 0  ω   − slip − Lr − − − slip − Lr B = 1 I and sign(B) = I . σLs 2 2 Energies 2020, 13, 1276 7 of 15

In what follows, the adaptive controller is proposed.

Theorem 1. Let us consider the nonlinear system (7) with unknown parameters. The following adaptive controller is u = θ(Γ)Tω(K) n, (8) ∈ < n n guarantees that lim (y∗ y) = 0. Here, y∗ is the known required output, fixed by the operator, u t − ∈ < ∈ < →∞ is the controller (8) output, applied to the system (7) input, and the controller parameters K +(n n) and ∈ < × Γ (n+m) (n+m) are adjusted by the designer. ∈ < × Here, ω (n+m) is the accessible information vector given by ∈ <  T T  . T ω = f (y) Ke + y∗ , (9) where K n n is a Hurwitz matrix chosen by the designer as K = 1 with τ the known time constant of ∈ < × nτ system (7), and θT n (n+m) are the adaptive controller parameters adjusted through the adaptive laws ∈ < × . θ = sign(B)Te(Γω)T, (10)

Here, the fixed-gains Γ (n+m) (n+m) is a diagonal matrix computed according to the following normalized ∈ < × FG equation αI(n+m) Γ =  , (11) + T 1 ωratedωrated with ω (n+m) is the rated information vector defining the operational range of ω, α is a positive adjustable rated ∈ < parameter chosen by the designer, and I(n+m) is the identity matrix of order (n + m).

. Proof: Let us define the control error as e = y y. Subtracting y∗ to both sides of (7), adding ∗ − and subtracting the term Ke in the right-hand side, and making some algebraic arrangements, it is obtained that .  T  e = Ke + B θ∗ ω u (12) − − h i where θ T = B 1AB 1 n (n+m) is the controller ideal fixed parameters of the controller, ∗ − − ∈ < × which are unknown. Substituting (8) in (12), defining the controller parameters error as eθ = θ∗ θ . . − (which implies e = θ as θ is constant), and considering (9), the equations describing the evolution θ − ∗ of the errors are . . e = Ke + BeTωe = sign(B)Te(Γω)T (13) − i θ θ − Let us propose the following Lyapunov candidate function, which is positive definite   1 T 1 T T 1 V(e, e ) = e e + Trace e B e Γ− (14) θ 2 2 θ| | θ . T .  T T . 1 Taking the time derivative of (14), it is obtained that V(e, eθ) = e e + Trace e B eθΓ− . θ. | | Substituting into this last expression the derivatives of the controlled variable error e from (13) . T T T T 1 and the controller parameters error eθ (13), considering that (Γω) = ω Γ , Γ Γ− = 1 due to Γ is a diagonal matrix, that B equals the sign(B) multiplied by B , and grouping terms, we obtain . T T T  T T T | | V(e, eθ) = e Ke + e Be ω + Trace e B eω . Now, considering the two vectors property where −  θ − θ   aTb = Trace baT to write into the trace the second term eTBeTω = Trace eTωeTB and rearranging, we .   θ θ have V(e, e ) = eTKe + Trace eTBTeωT + eTBTeωT . Canceling terms allows obtaining θ − − θ θ . V(e, e ) = eTKe 0 (15) θ − ≤ Energies 2020, 13, 1276 8 of 15

As can be seen in (15), the first-time derivative of the Lyapunov function (14) is negative semidefinite; thus, using Lapunov Theorem, it can be concluded that system (13) is stable.

(a) Let us now prove that control error e converges to zero.

Integrating Equation (15) in the interval (0, ) it is obtained that ∞ Z ∞  V( ) V(0) = eTKe dλ (16) ∞ − − 0

Since the system is stable, e, e are all bounded (e, e ), and from (14) it can be concluded θ θ ∈ L∞ that V is always bounded. Thus, the left-hand side of (16) is always bounded, which implies from the right-hand side of (16) that e 2. ∈ L Since e and the reference output y is assumed to be bounded, it can be concluded from ∈ L∞ ∗ the errors expressions that y ∞. Thus, since f (y) is assumed a bounded function for a bounded y, ∈ L . . from (9) ω ∞, and consequently, from (13), it can be concluded that e, eθ ∞. ∈ L . ∈ L Thus, since e, e and e 2, using Barbalat’s Lemma [18] it can be concluded that ∈ L∞ ∈ L lime = 0 t →∞ and this concludes the proof. The controller parameters error eθ is stable, and the adaptive controller parameters θ not necessarily tend to the controller ideal parameters θ∗. 

The normalized FG-APBC for the IM is described in Figure3, where summarizing, u = θTω,  T T h iT T T p T  .  8 2 2 ω = y ω y ω y Ke + y∗ , with y = Ird Irq , e = (y∗ y) e 2 r ∈ < ∈ < − ∈ < h iT . with y = I 0 , θ = e(Γω)T due to sign(B)T = 1, and Γ = I8 with α = 1 and ∗ s_rated + T (1 ωratedωrated) h p iT ωrated = Is_rated 0 2π fratedIs_rated 0 2 ωr_ratedIs_rated 0 0 0 . The value K = 50/Jm is 50 1 chosen, with Jm the motor inertia taken from the IM datasheet. The value K = = came Jm (Jm/(10 5) from the widely used criterion considering the inner current loop 10 as times faster than× the outer mechanical loop and our proposal for the normalized FG-APBC of considering a controller time as a constant of a fifth the inner time constant of the plant. In order to verify the achievement of the proposal goal, comparative experimental results considering real-time simulations are described in the next section. Energies 2020, 13, x FOR PEER REVIEW 8 of 14

The normalized FG-APBC for the IM is described in Figure 3, where summarizing, 𝑢=𝜃𝜔, 𝜔=𝑦 𝜔 𝑦 𝜔 𝑦 (𝐾𝑒 +𝑦∗) ∈ℜ 𝑦=𝐼 𝐼 ∈ℜ 𝑒=(𝑦∗ −𝑦) ∈ℜ , with , with ∗ 𝑦 = 𝐼_ 0 , 𝜃 =𝑒(𝛤𝜔) due to 𝑠𝑖𝑔𝑛(𝐵) =1, and 𝛤= with 𝛼 = 1 and 𝜔 =𝐼 02𝜋𝑓 𝐼 0 𝜔 𝐼 0 00 𝐾 = 50/𝐽 _ _ _ _ . The value is chosen, with 𝐽 the motor inertia taken from the IM datasheet. The value 𝐾= = came (/( × ) from the widely used criterion considering the inner current loop 10 as times faster than the outer mechanicalEnergies 2020, 13 loop, 1276 and our proposal for the normalized FG-APBC of considering a controller time9 as of 15a constant of a fifth the inner time constant of the plant.

Figure 3. SimulinkSimulink block block diagram diagram of the normalized FG-APBC for the IM. 4. Experimental Comparative Results and Discussion In order to verify the achievement of the proposal goal, comparative experimental results consideringThe schemes real-time from simulations Figure1 (basic are described SCS) and Figurein the 2next (proposed section. HST-SCS) were applied to an IM of 200 HP. These were programmed in Simulink version 8.9 of Matlab R2017a (9.2.0.538062) for Win64. 4.Later, Experimental they were downloadedComparative into Results a real-time and Discussion simulator controller 4510 v2 from OPAL-RT and run. The field-programmable gate array (FPGA) of the OPAL was used to creates a hardware implementation of The schemes from Figure 1 (basic SCS) and Figure 2 (proposed HST-SCS) were applied to an IM portions of the software application. of 200 HP. These were programmed in Simulink version 8.9 of Matlab R2017a (9.2.0.538062) for The real-time simulations started at 0 s and stopped at 35 s using a sampling time of 10 µs. In both Win64. Later, they were downloaded into a real-time simulator controller 4510 v2 from OPAL-RT cases, a constant speed setpoint was applied equal to the nominal speed value and a ramp of 50 rpm/s. and run. The field-programmable gate array (FPGA) of the OPAL was used to creates a hardware The inverter trip unit was running at the FPGA of the controller and works at 2 kHz. The motor implementation of portions of the software application. data plate is 149.2 kW, 460 V, 60 Hz, 1755 rpm (183.8 rad/s), fp = 0.85, η = 86.6%, T = 812 Nm, and the The real-time simulations started at 0 s and stopped at 35 s using a samplingnom time of 10 μs. In motor inertia taken from its datasheet is 3.1 Kgm2. The parameters of the motor-load set (considered both cases, a constant speed setpoint was applied equal to the nominal speed value and a ramp of 50 unknown) are p = 4, L = 10.46 mH, L = L = 10.7627 mH,R = 14.85 mΩ, R = 9.295 mΩ, J = 6.2 Kgm2, rpm/s. m s r s r and B = 0.08 Nms. The inverter trip unit was running at the FPGA of the controller and works at 2 kHz. The motor data4.1. Basicplate SCS is 149.2 kW, 460 V, 60 Hz, 1755 rpm (183.8 rad/s), fp = 0.85, η = 86.6%, Tnom = 812 Nm, and the motor inertia taken from its datasheet is 3.1 Kgm2. The parameters of the motor-load set √ In this case, the following SCS configuration parameters were used: Vs_rated = 460/ 3, frated = 60 Hz, fc = 40% frated, Vboost = 15% Vs_rated, and fmin = 6% frated. The experimental results for a starting torque Tl = 30% Tnom, increasing to 110% Tnom at 5 s, are shown in Figure4. Energies 2020, 13, x FOR PEER REVIEW 9 of 14

(considered unknown) are p = 4, Lm = 10.46 mH, Ls = Lr = 10.7627 mH, Rs = 14.85 mΩ, Rr = 9.295 mΩ, J = 6.2 Kgm2, and B = 0.08 Nms.

4.1. Basic SCS

In this case, the following SCS configuration parameters were used: 𝑉_ = 460/√3, 𝑓 = Energies60 Hz, 2020𝑓 =, 13 40%, 1276 𝑓, 𝑉 = 15% 𝑉_, and 𝑓 =6% 𝑓 . The experimental results10 for of 15a starting torque Tl = 30% Tnom, increasing to 110% Tnom at 5 s, are shown in Figure 4.

Figure 4. Real-time simulation results of the basic SCS for a 200 HP IM.

Figure4 4 shows shows the the results results of of the the basic basic SCS, SCS, starting starting with with 30% 30% of theof the nominal nominal torque. torque. The The slope slope of theof the voltage voltage amplitude amplitude decreased decreased at a timeat a time between between 15 and 15 20and s at 20 40% s at of 40% the requiredof the required speed. speed. A torque A rippletorque isripple shown is atshown low speeds.at low speeds. The real The rotor real speed rotor followed speed followed the required the required rotor with rotor an accuracywith an ofaccuracy 2.2%. of 2.2%.

There is a smooth change when switching the voltage in Equation (1) at the the frequency frequency 𝑓fc.. Neither the rotorrotor speed,speed, thethe statorstator current,current, nornor thethe torquetorque isis aaffectedffected byby thisthis changechange ofofthe thecontrol controlmethod. method.

4.2. Proposed Adaptive HST-SCS V = The SCS configurationconfiguration parameters in this case were the same used by the basic SCS (𝑉_s_rated = 460/ √3, f = 60 Hz, f = 40% f , V = 15% V , and f = 1% f ), and additionally, 460/√3 , rated𝑓 =60 Hzc, 𝑓 = 40%rated 𝑓boost, 𝑉 =s 15%_rated 𝑉_min, and 𝑓rated=1% 𝑓 ), and Iv 2 fc1 = 8% frated, Is_rated = 255 A, Γ = T , and Jm = 3.1 kgm . The experimental results are additionally, 𝑓 =8% 𝑓 , 𝐼_(1+ω= 255ωrated 𝐴) , 𝛤= , and 𝐽 =3.1 kgm . The rated shown in Figure5. experimental results are shown in Figure 5. Results from Figure5 show the achievement of the proposed goal after starting with 110% of the nominal torque. Conversely to the basic SCS results from Figure4, the HST-SCS is characterized by having the motor magnetized from zero speed consuming the rated stator current. It has a higher voltage slope after starting. It also consumes a starting stator current smaller than the basic SCS loaded with 30% of the nominal torque (see Figure4). After fc1, the HST-SCS behaves like the basic SCS: the slope of the voltage amplitude decreased at a time between 15 to 20 s at 40% of the required speed; a torque ripple is also shown at low speeds, and the real rotor speed followed the required rotor with an accurate of 2.2%. Distinctly from the basic SCS (Figure4), the HST-SCS (Figure5) shows an extra change in the slope of the voltage amplitude. It is higher at a time between 3 and 4 s at 8% of the required speed. At this point, the control strategy changes from the HST-SCS to the SCS, showing higher torque amplitude in correspondence to the higher torque load. The exact switching point fixed at 8% could change, looking for a smoother strategy change.

Energies 2020, 13, 1276 11 of 15 Energies 2020, 13, x FOR PEER REVIEW 10 of 14

FigureFigure 5.5. Real-time simulation resultsresults ofof thethe proposedproposed adaptiveadaptive HST-SCSHST-SCS forfor aa 200200 HPHP IM.IM.

WhenResults switching from Figure between 5 show its the control achievement methods, of the the rotor proposed speed, goal the after stator starting current, with and 110% the torque of the shownnominal in torque. Figure 5Conversely are a ffected to inthe the basic HST-SCS. SCS results In particular, from Figure there 4, isthe a HST-SCS torque ripple is characterized caused by the by algorithmhaving the switching motor magnetized which is higher from zero than speed for the cons basicuming SCS (Figurethe rated4). Thisstator is current. a disadvantage It has a ofhigher the HST-SCSvoltage slope over theafter basic starting. SCS that It also should consumes be improved a starting in future stator work. current smaller than the basic SCS

loaded with 30% of the nominal torque (see Figure 4). After 𝑓, the HST-SCS behaves like the basic 5.SCS: Conclusions the slope of the voltage amplitude decreased at a time between 15 to 20 s at 40% of the required speed;An a HST-SCStorque ripple for IM is also was shown proposed at low based speeds, on the and basic the real SCS. rotor The proposedspeed followed SCSconsiders the required an adaptiverotor with current an accurate controller of 2.2%. based on a direct APBC scheme using a normalized FG to fine-tune the adaptiveDistinctly controller from parameter the basic thatSCS works(Figure without 4), the theHST-SCS knowledge (Figure of the5) shows motor-load an extra parameters. change in The the mainslopeadvantages of the voltage of the amplitude. HST-SCS It include is higher the at capability a time between to move 3 higher and 4 starting- at 8% of the required loads, together speed. withAt this a simple point, andthe cost-econtrolffective strategy implementation changes from without the HST-SCS needing to athe rotor SCS, speed showing sensor higher or estimator, torque variableamplitude observers, in correspondence or parameter to estimators. the higher Thetorque basic load. SCS The for exact IM and switching the HST-SCS point were fixed applied at 8% could to an IMchange, of 200 looking HP and for tested a smoother using the strategy real-time change. simulator controller 4510 v2 from OPAL-RT. The HST-SCS startedWhen with switching 110% of the between nominal its torque, control consuming methods, the less rotor stator speed, current the than stator the current, basic SCS and with the torque a load torqueshown of in 30% Figure of the 5 are nominal affected torque. in the Besides HST-SCS. the ratedIn particular, voltage perthere phase is a neededtorque byripple the basiccaused SCS, by thethe proposedalgorithm HST-SCS switching needs which the is rated higher current than for per the phase basic and SCS motor (Figure inertia 4). takenThis is from a disadvantage the IM datasheet. of the HST-SCSThere over is a torquethe basic ripple SCS causedthat should by the be algorithm-switching improved in future ofwork. the HST-SCS, which is higher than for the basic SCS starting. This is a disadvantage of the HST-SCS over the basic SCS, which has a smooth5. Conclusions change between its methods. Neither the rotor speed, the stator current, nor the torque is affectedAn byHST-SCS this change for IM of thewas control proposed method based in theon basicthe basic SCS. SCS. The proposed SCS considers an adaptiveIn future current work, controller the proposed based on HST-SCS a direct should APBC bescheme validated using under a normalized a small scaleFG to test fine-tune bench. the In addition,adaptive controller using a motor parameter of low that rated works power, without the the torque knowledge ripple magnitude,of the motor-load increased parameters. during The the switchingmain advantages between of methods, the HST-SCS should beinclude studied the and capa decreasedbility to as move much higher as possible. starting-torque loads, together with a simple and cost-effective implementation without needing a rotor speed sensor or Author Contributions: Conceptualization and methodology, J.C.T.-T.; validation and software, C.C.-J.; writing—originalestimator, variable draft observers, preparation or parameter and visualization, estimators. J.C.T.-T., The N.A.-C.basic SCS and for A. IMÁ.-G.; and investigation, the HST-SCS formal were analysis,applied writing—reviewto an IM of 200 and HP editing, and tested all authors; using supervision,the real-time M.V.-L. simulator and Á .C.-M.;controll projecter 4510 administration, v2 from OPAL- data curation,RT. The resourcesHST-SCS and started funding with acquisition, 110% of the J.C.T.-T. nominal and M.A.D.-M.torque, consuming All authors less have stator read andcurrent agreed than to thethe publishedbasic SCS version with a ofload the torque manuscript. of 30% of the nominal torque. Besides the rated voltage per phase needed Funding:by the basicThis SCS, research the was proposed funded byHST-SCS FONDEF needs Chile, th grante rated ID17I10338; current FONDECYT per phase Chile,and motor grants inertia 11170154 taken and 1190959;from the and IM CONICYT datasheet. Chile Project AFB180004. ConflictsThere of Interest:is a torqueThe ripple authors caused declare by no the conflict algorithm- of interest.switching of the HST-SCS, which is higher than for the basic SCS starting. This is a disadvantage of the HST-SCS over the basic SCS, which has a

Energies 2020, 13, x FOR PEER REVIEW 11 of 14 smooth change between its methods. Neither the rotor speed, the stator current, nor the torque is affected by this change of the control method in the basic SCS. In future work, the proposed HST-SCS should be validated under a small scale test bench. In addition, using a motor of low rated power, the torque ripple magnitude, increased during the switching between methods, should be studied and decreased as much as possible.

Author Contributions: Conceptualization and methodology, J.C.T.-T.; validation and software, C.C.; writing— original draft preparation and visualization, J.C.T.-T., N.A.-C. and A.A.-G.; investigation, formal analysis, writing—review and editing, all authors; supervision, M.V.-L. and Á.C.-M.; project administration, data curation, resources and funding acquisition, J.C.T.-T. and M.A.D.-M.

Funding: This research was funded by FONDEF Chile, grant ID17I10338; FONDECYT Chile, grants 11170154 and 1190959; and CONICYT Chile Project AFB180004.

EnergiesConflicts2020 of, 13Interest:, 1276 The authors declare no conflict of interest. 12 of 15

Appendix A

Appendix A.1A.1. Steady-StateSteady-State IMIM ModelModel perper PhasePhase The steady-state IMIM modelmodel perper phase phase of of Figure Figure A1 A1[ 3 [3]] allows allows modeling modeling the the steady-state steady-state behavior behavior of of the IM. The variables are as follows: the stator voltage per phase, 𝑣 ; the stator current per phase, the IM. The variables are as follows: the stator voltage per phase, vs; the stator current per phase, is; the 𝑖 ; the rotor current per phase, 𝑖 (which is assumed to be a sinusoidal signal); the rotor angular rotor current per phase, ir (which is assumed to be a sinusoidal signal); the rotor angular speed at the speed at the shaft, 𝜔 ; the electrical angular frequency, 𝜔 ; the slip angular frequency of the rotor,p  shaft, ωr; the electrical angular frequency, ωe; the slip angular frequency of the rotor, ω = ωe ωr ; slip 2 𝜔 =𝜔 − 𝜔 ; the poles pair, p; the electromagnetic motor output torque, 𝑇 ; and the− load the poles pair, p; the electromagnetic motor output torque, Tem; and the load torque, Tl. Rs, Rr are the statortorque, and 𝑇 rotor. 𝑅,𝑅 resistances are the ofstator a phase and winding rotor resistances respectively; ofand a phaseLs, Lr , windingand Lm are respectively; the stator, rotor,and and𝐿,𝐿 magnetizing,and 𝐿 are inductances,the stator, rotor, respectively. and magnetizing inductances, respectively.

Figure A1. Steady-state IM model per phase. (a) Power flow diagram, (b) components and parts, Figure A1. Steady-state IM model per phase. (a) Power flow diagram, (b) components and parts, (c) (c) equivalent circuit. equivalent circuit.

The active power flow is shown in Figure A1a. The electrical input power (Pin = 3 e( vs is ) 2 R | || | consumed by the motor is lost by the stator winding heating (Ps = 3 is Rs) by heating of the rotor 2 | | cage (Pr = 3 ir Rr), obtaining the converted power (Pconv = Temωr) that, after the power losses (P ), | | losses finally delivers the mechanical power at the shaft to move the load (Pout = Tlωr) in correspondence with the IM parts of Figure A1b. Kirchhoff’s voltage law equation for the stator gives

, , vs = (Rr + jωeL )is + jωeLm(is ir) = (Rr + jωeL )is + jωeLmim, with im = (is ir) (A1) r − r − Energies 2020, 13, 1276 13 of 15

Moreover, the electromechanical torque Tem is given by the ratio between the power converted 2 ωr   3 ir Rr ω p Pconv | | slip p 2 Rr Pconv and the angular speed of the rotor ωr , i.e., Tem = p = p = 3 ir . Using 2 2 ωslip 2 ωr 2 ωr | | the expression jLm ir = is, (A2) Rr + jLr ωslip obtained from the Kirchhoff’s current law equation for the rotor   p   2 ωr jωeLmis + jωe(Lr0 + Lm)ir + Rr 1 + ir = 0 , we get − ωslip

2

p jLm Rr 2 Tem = 3 is . (A3) Rr 2 + jLr ωslip | | ωslip

1 Substituting into (A3) the expression given by is = Zeq− us, together with the IM equivalence       ωsync  ,  impedance obtained as Zeq = Rs + jωsyncL + jωsyncLm jωsyncL + Rr = Rs + jωsyncL + s0 r0 ωslip s   k ωsync jωsyncLm jωsyncLr0+Rr ωslip (RsRr ωeωslipLrLs0 ωeωslipLmLr0)+j(ωslipRsLr+ωeRrLs0+RrLmωe) ωsync , which is equal to Zeq = − − , jωsyncLr+Rr (Rr+jωslipLr) ωslip and rearranging terms, we finally have

2 p ωslipLm T = 3 R V 2 . (A4) em r s  2  2 2 2 RsRr ω ωeLsLr + ω ωeLm + ω RsLr + ωeRrLs − slip slip slip Appendix A.2 IM Dynamic Model The IM dynamical model is obtained under the assumptions given in [20] (Section 2.1). These assumptions are as follows: the rotor and stator windings are distributed symmetrically; the signals are sinusoidal (neglecting the harmonic effects); hysteresis, iron losses, and saturation are negligible; the IM is working into the linear zone; all motor parameters are constant and referred to the stator; a two-pole machine is considered with results expandable to more poles; and a quadrature-phase machines smooth-air-gap is considered. Applying Kirchhoff laws for the stator and rotor circuit [20], using the Park transformation [22] to express the electrical equations into a rotating synchronous reference frame, and splitting the vectors into real and imaginary parts, the IM d-q dynamical model used for the HST-SCS scheme is obtained.

. Rs0 RrLm Lm p 1 I = I + ωeIsq + Ψ ωrΨrq + V sd σLs sd σL L2 rd σLsLr 2 σLs sd . − s r − Rs0 RrLm Lm p 1 Isq = Isq ωeI + Ψrq + ωrΨ + Vsq σLs sd σL L2 σLsLr 2 rd σLs − . − s r  (A5) Rr p RrLm Ψrd = Ψrd + ωe ωr Ψrq + Isd . − Lr  − 2  Lr Rr p RrLm Ψrq = Ψrq ωe ωr Ψ + Isq − Lr − − 2 rd Lr where the variables are the amplitude of the sinusoidal signals at the motor terminals expressed as the direct and quadrature stator current amplitudes Isd and Isq, respectively; the direct and quadrature stator voltage amplitudes are Vsd and Vsq, respectively, and the direct and quadrature rotor flux amplitudes are Ψrd and Ψrq, respectively. ωr is the rotor angular speed at the shaft, and ωe is the angular electrical frequency or speed of the synchronous reference frame. The parameters Rs, Rr are the stator and rotor resistances, respectively, of a phase winding; p is the poles number, Ls, Lr, and Lm are the stator, rotor, 2 2 Lm LmRr and magnetizing inductances, respectively, σ = 1 L L is the dispersion coefficient, and Rs0 = Rs + 2 − r s Lr  p  is the stator transient resistance and the slip angular frequency of the rotor ω = ωe ωr . slip − 2 Energies 2020, 13, 1276 14 of 15

The coupling between electromagnetic torque, stator current, and rotor flux can be observed  p  in Equation (A5) including the nonlinear terms, such as ωgIsd, ωgIsq, ωrΨrq, ωrΨrd, ωg 2 ωr Ψrq,  p  − ωg ωr Ψ , Ψ Isq, and ΨrqI . − 2 rd rd sd

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